New Directions in Geometric Group Theory and Topology

该项目的目标是对各级数学产生最大的影响。在研究层面，该项目为我们对四维空间的理解引入了新的几何观点。每当4个参数出现时，这些空间就会出现，例如跟踪飞机的三维位置及其在该位置的时间。该项目将开发新的方法来理解这些空间的对称性，从“K3流形”开始，这是物理学中出现的一个基本例子。该项目还将培养数名博士生；继续PI在媒体上的公众宣传；并通过继续PI与数学联盟的合作，为数学的多样性做出贡献。PI建议将瑟斯顿观点引入到K3曲面的映射类群的研究中。具体目标包括：瑟斯顿三分法；瑟斯顿范式；尼尔森(非)实现;新的美丽的例子/结构。在第二个方向上，PI提出了一个指导原则，由代数几何的神奇构造激发，在几何群论、拓扑和代数几何中产生了许多自然的、激发的问题和猜想，涉及许多不同模空间的各种形式的刚性。PI还将培训研究生并开展外联活动，作为更广泛影响的一部分。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The goal of this project is to have maximal impact on mathematics at all levels. At the research level, the project introduces new geometric points of view to our understanding of 4-dimensional spaces. These spaces arise whenever 4 parameters arise, for example keeping track of the 3-dimensional position of an airplane together with the time it is at that position. The project will develop new methods to understand symmetries of these spaces, starting with ``K3 manifolds'', a fundamental example arising in physics. The project will also lead to the training of several PhD students; continue the PI's public outreach in the media; and contribute to work for diversity in mathematics by continuing the PI's work with the Math Alliance.The PI proposes to introduce the Thurstonian point-of-view into the study of mapping class groups of K3 surfaces. Specific goals include: Thurston trichotomy; Thurston normal form; Nielsen (non)realization; new beautiful examples/constructions. In a second direction, the PI proposes a guiding principle, motivated by the miraculous constructions of algebraic geometry, that produces many natural, motivated problems and conjectures in geometric group theory, topology and algebraic geometry, concerning various forms of rigidity for many different moduli spaces. The PI will also train graduate students and conduct outreach as part of broader impacts.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.